Ξ WENO 34 1 Vol 34,No 1 2 0 0 4 2 JOURNAL OF UNIVERSITY OF SCIENCE AND TECHNOLOGY OF CHINA Feb 2 0 0 4 :025322778 (2004) 0120029209,, (, 230026) :Weighted Essentially Non2Oscillatoy (WENO) WENO,,, :WENO ; ;Runge2Kutta :V211 1 :A AMS Subject classif ication( 2000) : 65M06,65M99,35L65 1 0,,, 90, ENO ( Essentially Non2Oscillatoy) [4,5 ] WENO (Weighted ENO) WENO,,, Liu, Oshe Chan [12 ] WENO,Jiang,Shu, Hu Qiu [1,3,6,7,14,16 WENO ] WENO, CPU, Levy Puppo Russo [10 ] TVD2Runge2 Kutta,, WENO, [8,9,13 ], WENO,, 1 WENO u t + f ( u) x = 0 (1) Ξ :2002204220 : (10071083) :,,1978, E2mail :xuzl @mail ustc edu cn 1995-2005 Tsinghua Tongfang Optical Disc Co, Ltd All ights eseved
30 34 [7 WENO ] x = x j, f ( u) x x = x j 1 x ( f^ j +1/ 2 - f^j - 1/ 2), f^j +1/ 2 WENO f ( u) 0, k, S ( j) = { x j -, x j - +1,, x j - + }, = 0,1,,, (2) f^j +1/ 2, k, j +1/ 2 = p ( ) ( x j +1/ 2 ) = f^ ( ) i = 0 c i f ( u j - + i ), = 0,1,,, (3) p ( ) ( x) S WENO f^ j +1/ 2 = i = 0 f^( ) j +1/ 2, (4),, 0, = 1 i = 0 f ( u), d, (Linea Weights), f^ j +1/ 2 = d i = 0 f^( ) j +1/ 2 (5), f^j +1/ 2 2 d, T = 2 P( x), d k = 3, = 0 S P( x j +1/ 2 ) = d p ( ) ( x j +1/ 2 ) (6) i = 0 d 0 = 1 10, d 1 = 3 5, d 2 = 3 10 (7) P( x) 2,,, Jiang Shu, = h l = 1 2 l - 1 Ij d l d x l p ( ) ( x) 2 d x (8), p ( ) ( x) I j = ( x j - 1/ 2, x j +1/ 2 ) S ( j) k = 3, 0 = 13 12 ( f j - 2-2 f j - 1 + f j ) 2 + 1 4 ( f j - 2-4 f j - 1 + 3 f j ) 2, 1 = 13 12 ( f j - 1-2 f j + f j +1 ) 2 + 1 4 ( f j - 1 - f j +1 ) 2, 2 = 13 12 ( f j - 2 f j +1 + f j +2 ) 2 + 1 4 (3 f j - 4 f j +1 + f j +2 ) 2, = s = 0 s, = d ( + ) 2, = 0,1,,, (9) 1995-2005 Tsinghua Tongfang Optical Disc Co, Ltd All ights eseved
1 WENO 31, = 10-6 : ( ), d, 2 ; ( ),WENO ENO, f ( u) 0,, f ( u) 0 WENO f ( u), f ( u) = f + ( u) + f - ( u), ( f + ) ( u) 0, ( f - ) ( u) 0 WENO f + ( u) f - Lax2Fiedichs, ( u), f ( u) = 1 2 [ f ( u) u ], = max u f ( u) (10), f ( u) Jacobi [7 ] : = d k + O ( h ), = 0,1,,, WENO 2 = D (1 + O ( h ) ), D Taylo TVD2Runge2Kutta u (1) = u n + t L ( u n ), u (2) = 3 4 u n + 1 4 u (1) + 1 4 t L ( u (1) ), u n +1 = 1 3 u n + 2 3 u (2) + 2 3 t L ( u (1) ) ( l) ( l) Runge2Kutta,, ( l) =, ( l) =, k = 0,1,2, l = 1,2, Eule, ( l) = = D (1 + O ( h ) ),,,WENO ENO, k = 3, 1, / 1 0, = 0,2, t c t 1, c CFL, TVD2Runge2Kutta = f ( u (1) i ), Taylo, f i (1) G l ( u) f i (1) = f i n + t G 1 ( u i n ) + ( t 2 2) G 2 ( u i n ) + ( t 3 3!) G 3 ( u i n ) +, = 5t l f ( u) L ( u) Taylo (1) = (1 + O ( h) ), = 1,2,3 (1) WENO, Taylo 1995-2005 Tsinghua Tongfang Optical Disc Co, Ltd All ights eseved
32 34 t = O ( h), (1) / 1 (1) 0, = 0,2 (2) / 1 (2) 0, = 0,2,, WENO ( ),, ( ),, Roe,,WENO, 2 WENO, TVD2Runge2Kutta, WENO 1, u t + u x = 0, (11) u ( x,0) = sin (2 x), T = 2 1 WENO 2 Buges u t + ( u 2 2) x = 0, (12) u ( x,0) = 0 5 + sin ( x), t = 0 5/, Buges, 2 t = 1 5/, 1 t = 0 5/, 1995-2005 Tsinghua Tongfang Optical Disc Co, Ltd All ights eseved
1 WENO 33 1 Buges Fig 1 Dimensional linea scala quantity Buges equations 3 Eule v v E t + v 2 + p v ( E + p) x = 0, (13) p : v p, E = - 1 + 1 2 v 2, = 1 4 Lax (, v, p) (, v, p) Sod (, v, p) (, v, p) = (0 445,0 698,3 528), x 0 = (0 5,0,3 571), x > 0, = (1 0,0,1 0), x 0 = (0 125,0,0 1), x > 0, 1 6 2 0, 2, 2 Eule Riemann Fig 2 The Riemann value poblem fo Euleian equations 4 Eule, (, v, p) = (3 857 143,2 629 369,10 333 333), x < 4 (, v, p) = (1 + sin (5 x),0,1), x 0 1995-2005 Tsinghua Tongfang Optical Disc Co, Ltd All ights eseved
34 34 = 0 2,,, t = 1 8 3, 2000 Runge2Kutta WENO, N = 400 3 Eule Fig 3 The collision poblem between shock and density wave fo Euleian equation 5 Eule ( l, v l, p l ) = (1,0,1000), x 0 1, ( c, v c, p c ) (, v, p ) = (1,0,0 01), 0 1 < x < 0 9, = (1,0,100), x > 0 9 [0,1 ], 0 038, [7, ], CFL, CFL,,, [16, ] CFL (CFL = 0 6), 4, 6 Buges u t + u2 2 x u ( x, y,0) = 0 5 + sin ( x + y) / 2 + u2 2 y = 0, (14), [0,4 ] [0,4 ] t = 1 5/ 5 7 Mach 10, 60, = 1 4, p = 1 0 [0,4 ] [0,1 ], 1 6 < x < 4, 0 < x < 1 6,, Mach 10, 1995-2005 Tsinghua Tongfang Optical Disc Co, Ltd All ights eseved
1 WENO 35 4 Eule Fig 4 Inteacting stong blast waves of Euleian equation 5 Buges,80 80 Fig 5 22dimensioal Buges equations, t = 0 2 6 7, 30 : ( ), 2 G CPU 512M IBM, Com2 paq Visual Fotan 6 5, Lax, WENO 20 25 %, 6 1 5 22 7, 6 % 30, Fig 6 Double Mach eflection paoblem ; Dansity ;, contou lines fom = 1 5 to = 22 7 1995-2005 Tsinghua Tongfang Optical Disc Co, Ltd All ights eseved
36 34 7 1 5 22 7 30 Fig 7 Double Mach eflection poblem Blon2up eqion aound the double Mach system Density ; 30 contou lines fom = 1 5 to = 22 7, ( ) WENO CFL, Couant, WENO CFL 8 Lax Fig 8 Lax shock wave pipe poblem 1995-2005 Tsinghua Tongfang Optical Disc Co, Ltd All ights eseved
1 WENO 37 Gibbs,,,,,, Lax TVD2Runge2Kutta, WENO ( 8 ),,,, WENO, Runge2Kutta, 8,, [ 1 ] Balsaa D and Shu C W Monotonicity pesev2 ing weighted essentially non2oscillatoy schemes with inceasingly high ode accuacy [ J ] J Comput Phys,2000,160 : 4052452 [2 ] Deng X and Zhang H Developing high2ode weighted compact nonlinea schemes J Com2 put Phys,2000,165 : 22244 [3 ] Fiedichs O Weighted essentially non2oscilla2 toy schemes fo the intepolation of mean val2 ues on unstucted gids [ J ] J Comput Phys,1998,144 : 1942212 [4 ] Haten A, Engquist B,Oshe S and Chakaa2 vathy S Unifomly high ode accuate essen2 tially non2oscillatoy schemes [J ] J Com2 put Phys,1987,71 : 2312303 [5 ] Haten A and Oshe S Unifomly high ode accuate essentially non2oscillatoy schemes, I, SIAM J Nume Ana,1987,24 : 2792309 [ 6 ] Hu C And Shu C W Weighted essentially non2 oscillatoy schemes on tiangula meshes[j ] J Comput Phys,1999,150 : 972127 [7 ] Jiang G S and Shu C W Efficient implementa2 tion of weighted ENO schemes[j ] J Comput Phys,1996,126 : 2022228 [8 ] Levy D,Puppo G and Russo G On the behav2 io of the total vaiation in CWENO methods fo consevation laws [ J ] Appl Nume Math,2000,33 : 4072414 [9 ] Levy D, Puppo G and Russo G A thid ode cental WENO scheme fo 2D consevation laws [ J ] Appl Nume Math, 2000, 33 : 4152421 [10 ] Levy D,Puppo G and Russo G Cental WENO schemes fo hypebolic systems of consevation laws[j ] Math Model Nume Anal,1999, 33 : 5472571 [11 ],,, [J ],2001,31 : 2452262 [ 12 ] Liu X D,Oshe S and Chan T Weighted essen2 tially non2oscillatoy schemes [ J ] J Comput Phys,1994,115 : 2002212 [13 ] Qiu J X and Shu C W On the constuction, compaison and local chaacteistic decomposi2 tion fo high ode cental WENO schemes[j ] Submit to J Comput Phys ( 54 ) 1995-2005 Tsinghua Tongfang Optical Disc Co, Ltd All ights eseved
54 34 FB G and t ansvese loading is obtained expeimentally A multi2point dist ibute t ansvese load sensing by means of wavelengt h addessing is also investigated wit h an expeiment of two2gat2 ing simultaneous sensing Key wods : fibe Bagg gating ; tansvese load ; stess induced biefingence ; esonance wave2 lengt h sepaation ; multi2point dist ibute sensing ( 37 ) [ 14 ] Qiu J X and Shu C W Finite diffeence WENO schemes with Lax2Wendoff type time dis2 cetizations[j ] Submit to SIAM J Sci Com2 put [15 ] Shi J, Hu C and Shu C W A technique of teating negative weights in WENO scheme [ R ] ICASE Repot No 2000249 [ 16 ] Shu C W Essentially non2oscillatoy and weighted essentially non2oscillatoy schemes fo hypebolic consevation laws [ R ] ICASE Re2 pot No 97265 [17 ] Wang Z J and Chen R F Optimized weighted essentially non2oscillatoy schemes fo linea waves with discontinuity [ J ] J Comput Phys,2001,174 : 3812404 Some Optimal Methods f o WENO Scheme in Hypabolic Consevation La ws XU Zhen2li, Q IU Jian2xian, L IU Ru2xun ( Depat ment of Maths, US TC, Hef ei, A nhui, 230026) Abstact : WENO (weighted Essentially Non2Oscillat oy) is a high2esolution numeical scheme used fo solving equations of hypebolic consevation laws In t his pape, some optimal st ate2 gies of the WENO scheme of hypebolic consevation laws ae discussed and the time of nonlin2 ea weighted computation and chaacteistic decomposing is educed By some numeical exam2 ples, the feasibility of these stategies is poved and the advantages and disadvantages com2 paed Key wods :WENO scheme ; hypebolic consevationa law equation ; Runge2Kutta met hod 1995-2005 Tsinghua Tongfang Optical Disc Co, Ltd All ights eseved